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Matterwave

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This has to do with the ADM formulation of GR. I am following MTW chapter 21 and Wald appendix E and chapter 10.

On page 510 in MTW, they are talking about the covariant derivative on the hypersurface defined via the covariant derivative on the 4-manifold. They take the 4-D covariant derivative of an arbitrary vector in equation 21.54 and then go on to comment on the fact that it has a component proportional to [itex]\mathbf{e}_0\cdot \mathbf{n}[/itex] which does not lie on the hypersurface, and they claim that to take the 3-D covariant derivative is to project away (kill in their language) this component so that we are left with a vector which lies entirely in the hypersurface. I understand them up until here.

What I don't understand is their next statement that says one can do this by simply changing the mu index which runs from 0 to 3 into an m index which runs from 1 to 3 in equation 21.54. Since the term which actually appears in 21.54 is a term that is proportional to [itex]\mathbf{e}_0[/itex], we should ONLY be able to project away the normal component of this. Given that [itex]\mathbf{e}_0=N\mathbf{n}+\mathbf{N}[/itex], we should be left with a term that is proportional to [itex]\mathbf{N}[/itex] shouldn't we? I don't see how we can just get rid of that term.

Later on in the same section, MTW discusses the covariant derivative of a oneform. In there, he seems to insist on using only the Christoffel symbols of the first kind (all lower indices), is there a reason for this? It seems natural to use the regular Christoffel symbols of the second kind (one upper and 2 lower indices)...it just seems a little weird that he would do this for no reason.

In this respect I follow Wald a little better because he explicitly uses projection operators [itex]h^a_b[/itex] instead of assuming a-priori the slicing of space-time that we have defined (so he wouldn't express the 3-D projection of 4-D objects in terms of lapse and shift, but simply in terms of this projection operator). What confuses me about him is that he seems to use this operator to project both covariant and contravariant indices. It makes sense to me to project contravariant indices because [itex]T\Sigma[/itex] (where [itex]\Sigma[/itex] is the hypersurface) is a vector subspace of TM (where M is our manifold), but it doesn't make sense to me to use this operator to project the covariant indices because [itex]T^*\Sigma[/itex] is not a vector subspace of T*M. In fact, one forms which live on T*M naturally induce one forms which live on [itex]T^*\Sigma[/itex] by simply limiting their action. Is he using the projection operator for covariant indices to simply mean this limiting of their action?

On page 510 in MTW, they are talking about the covariant derivative on the hypersurface defined via the covariant derivative on the 4-manifold. They take the 4-D covariant derivative of an arbitrary vector in equation 21.54 and then go on to comment on the fact that it has a component proportional to [itex]\mathbf{e}_0\cdot \mathbf{n}[/itex] which does not lie on the hypersurface, and they claim that to take the 3-D covariant derivative is to project away (kill in their language) this component so that we are left with a vector which lies entirely in the hypersurface. I understand them up until here.

What I don't understand is their next statement that says one can do this by simply changing the mu index which runs from 0 to 3 into an m index which runs from 1 to 3 in equation 21.54. Since the term which actually appears in 21.54 is a term that is proportional to [itex]\mathbf{e}_0[/itex], we should ONLY be able to project away the normal component of this. Given that [itex]\mathbf{e}_0=N\mathbf{n}+\mathbf{N}[/itex], we should be left with a term that is proportional to [itex]\mathbf{N}[/itex] shouldn't we? I don't see how we can just get rid of that term.

Later on in the same section, MTW discusses the covariant derivative of a oneform. In there, he seems to insist on using only the Christoffel symbols of the first kind (all lower indices), is there a reason for this? It seems natural to use the regular Christoffel symbols of the second kind (one upper and 2 lower indices)...it just seems a little weird that he would do this for no reason.

In this respect I follow Wald a little better because he explicitly uses projection operators [itex]h^a_b[/itex] instead of assuming a-priori the slicing of space-time that we have defined (so he wouldn't express the 3-D projection of 4-D objects in terms of lapse and shift, but simply in terms of this projection operator). What confuses me about him is that he seems to use this operator to project both covariant and contravariant indices. It makes sense to me to project contravariant indices because [itex]T\Sigma[/itex] (where [itex]\Sigma[/itex] is the hypersurface) is a vector subspace of TM (where M is our manifold), but it doesn't make sense to me to use this operator to project the covariant indices because [itex]T^*\Sigma[/itex] is not a vector subspace of T*M. In fact, one forms which live on T*M naturally induce one forms which live on [itex]T^*\Sigma[/itex] by simply limiting their action. Is he using the projection operator for covariant indices to simply mean this limiting of their action?

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